- Email: [email protected]
ONe-Way Deficit of SU(2) Invariant States
Vol. 73 (2014)
REPORTS ON MATHEMATICAL PHYSICS
No. 2
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES YAO -K UN WANG1,2 , T ENG M A1 , S HAO -M ING F EI1,3 and Z HI -X I WANG1 1 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China 2 College of Mathematics, Tonghua Normal University, Tonghua 134001, China 3 Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
(e-mails: [email protected] [email protected] [email protected] [email protected]) (Received June 12, 2013 – Revised November 12, 2013) We calculate analytically the one-way deficit for systems consisting of spin-j and spin- 12 subsystems with SU(2) symmetry. Comparing our results with the quantum discord of SU(2) invariant states, we show that the one-way deficit is equal to the quantum discord for half-integer j , and is larger than the quantum discord for integer j . Moreover, we also compare the one-way deficit with entanglement of formation. The quantum entanglement tends to zero if j increases, while the one-way deficit can remain significantly large. Keywords: one-way deficit, SU(2) invariant state, quantum discord.
1.
Introduction Quantum entanglement plays significant roles in the field of quantum information and quantum computation such as super-dense coding [1], teleportation [2], quantum cryptography [3], remote-state preparation [4]. Quantum correlations other than quantum entanglement also attract much attention recently [5–11]. Among them, quantum discord introduced by Oliver and Zurek and independently by Henderson and Vedral [6] is a widely accepted quantity. Quantum discord, a measure which quantifies discrepancy between the quantum mutual information and the maximal classical information, can be present in separable mixed quantum states. Following this discovery, much work has been done in investigating the properties and behaviour of quantum discord in various systems. Since complicated optimization procedure is involved in calculating quantum discord, one has no analytical formulae of quantum discord even for two-qubit quantum systems [12–17]. Other significant nonclassical correlations besides entanglement and quantum discord, for example, the quantum deficit [18, 19], measurement-induced disturbance [12], symmetric discord [20, 21], relative entropy of discord and dissonance [22], geometric discord [23, 24], and continuous-variable discord [25, 26] have been studied recently. The work deficit [18] proposed to characterize quantum correlations in terms of entropy production and work extraction by Oppenheim et al. is the first operational approach to connect the quantum correlations theory with quantum thermodynamics. Recently, Streltsov et al. [27, 28] has revealed that the one-way deficit plays [165]
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Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG
an important role in quantum correlations as a resource for the distribution of entanglement. The one-way deficit of a bipartite quantum state ρ is defined by [29]
k ρ k − S(ρ), (1) → (ρ) = min S { k }
k
where { k } are the projective measurements and S is the von Neumann entropy. The one-way deficit and quantum discord have similar minimum forms but they are different kinds of quantum correlations. We have obtained an analytical formula of one-way deficit for some well-known states such as Bell-diagonal states [30]. In this paper, we endeavoured to calculate the one-way deficit of bipartite SU(2) invariant states consisting of a spin-j and a spin- 12 subsystems. SU(2)-invariant density matrices of two spins S1 and S2 are defined to be invariant under U1 ⊗ U2 , that is, U1 ⊗ U2 ρU1† ⊗ U2† = ρ, where Ua = exp(i η · Sa ), a = 1, 2, are the usual rotation operator representations of SU(2) with real parameters η and h¯ = 1 [31, 32]. Those SU(2)-invariant states commute with all components of the total spin J = S1 + S2 . In real physical systems, SU(2)-invariant states arise from thermal equilibrium states of spin systems described by SU(2) invariant Hamiltonian [33]. The state space structure and entanglement of SO(3)-invariant bipartite quantum systems have been analyzed in the literature [34, 35]. For SU(2) invariant quantum spin systems composed of spin-S and spin- 12 , negativity is shown to be necessary and sufficient for separability [31, 32], and the relative entropy of entanglement has been analytically calculated [36] for (2j + 1) × 2 and ((2j + 1) × 3)-dimensional systems. Furthermore, the entanglement of formation (EoF), I-concurrence, I-tangle and convex-roof-extended negativity of the SU(2)-invariant states of a spin-j and spin- 12 [37] have been analytically calculated by using the approach of [38]. The quantum discord for SU(2)-invariant states composed of spin-j and spin- 21 systems has been analytically calculated in [39]. As an SU(2)-invariant state commutes with all components of J, ρ has the general form [31] S J 1 +S2 A(J ) ρ= |J, J z 00 J, J z | , (2) 2J + 1 J =|S1 −S2 | J z =−J where the constants A(J ) ≥ 0, J A(J ) = 1, |J, J z 0 denotes a state of the total spin J and the z-component J z . We consider the case with S1 of arbitrary length S and S2 of length 12 . Let S = j , J z = m. A general SU(2)-invariant density matrix has the form j − 12 1 F 1 ab ρ = j − 2 , m j − 2 , m 2j 1 m=−j + 2
1−F + 2(j + 1)
j + 12
j + 1 , m j + 1 , m , 2 2 1
m=−j − 2
(3)
167
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES
where F ∈ [0, 1], and F is a function of temperature in thermal equilibrium. ρ ab is a ((2j +1)×2)-bipartite state. It has two eigenvalues λ1 = F /2j and λ2 = (1−F )/(2j + 2) with degeneracies 2j and 2j +2, respectively [39]. The entropy of ρ ab is given by S(ρ ab ) = −F log
1−F F − (1 − F ) log . 2j 2j + 2
(4)
As the eigenstates of the total spin can be given by the Clebsch–Gordan coefficients [40] in the coupling of a spin-j to spin- 12 , j ± 1 , m 2 1 1 1 j + 12 ± m j + 12 ∓ m 1 1 j, m − , j, m + , − 1 , (5) ⊗ + ⊗ =± 2 2 2 2 2j + 1 2 2j + 1 2 the density matrix (3) can be written in the product basis form [39], F ρ ab = 2j
j − 12
2 a−
m=−j + 12
m − 1 m − 2
1 1 1 ⊗ 2 2 2
1 1 1 1 1 m+ ⊗ − + m + m− +a− b− m − 2 2 2 2 2 1 1 1 1 2 m + ⊗ − − m + +b− 2 2 2 2 j + 12 1−F 1 1 1 1 2 + m− ⊗ a+ m − 2(j + 1) 2 2 2 2 1
1 1 1 ⊗ − 2 2 2
1 1 1 1 1 m+ ⊗ − + m + m− +a+ b+ m − 2 2 2 2 2 1 1 1 1 2 m + ⊗ − − , +b+ m + 2 2 2 2
1 1 1 ⊗ − 2 2 2
m=−j − 2
where
j + 12 ± m a± = ± 2j + 1
and
b± =
(6)
j + 12 ∓ m . 2j + 1
When j = 12 , the state ρ ab turns out to be a 2 × 2 Werner state, I ρ = (1 − c) + c|ψ − ψ − |, 4 with |ψ − =
√1 (|01
2
− |10 ).
c=
4F − 1 , 3
(7)
168 2.
Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG
The main result Any von Neumann measurement on the spin- 12 subsystem can be written as Bk = V k V † ,
k = 0, 1,
(8)
where k = |k k|, |k is the computational basis, V = tI + i y · σ ∈ SU(2), σ = (σ1 , σ2 , σ3 ) with σ1 , σ2 , σ3 being the Pauli matrices, t ∈ R and y = (y1 , y2 , y3 ) ∈ R3 , t 2 + y12 + y22 + y32 = 1. Set j m(2Fj + F − j ) M= |m m| z3 j (j + 1)(2j + 1) m=−j √ j (j + 1) − m(m + 1)(2Fj + F − j ) +(z1 + iz2 ) |m m + 1| 2j (j + 1)(2j + 1) √ j (j + 1) − m(m + 1)(2Fj + F − j ) |m + 1 m| , +(z1 − iz2 ) 2j (j + 1)(2j + 1) where z1 = 2(−ty2 + y1 y3 ), z2 = 2(ty1 + y2 y3 ), z3 = t 2 + y32 − y12 − y22 with z12 + z22 + z32 = 1. After the measurement, one has an ensemble of post-measurement states {ρk , pk } with p0 = p1 = 12 and the corresponding post-measurement states [39] j 1 1 ρ0 = |m m| − M ⊗ V 0 V † = I − M ⊗ V 0 V † , (9) 2j + 1 m=−j 2j + 1 and
ρ1 =
j 1 1 † I + M ⊗ V 1 V † . (10) |m m| + M ⊗ V 1 V = 2j + 1 m=−j 2j + 1
The eigenvalues of ρ0 , ρ1 [39] are λ± n =
j −n 1 ± |(F (2j + 1) − j )|, 2j + 1 j (j + 1)(2j + 1)
(11)
where n = 0, . . . , j , and j denotes the largest integer that is less or equal to j . Obviously, eigenvalues are independent of the measurement. Analytical expression for quantum discord of ρ ab has been obtained in [39], 1−F F + (1 − F )log +1 2j 2j + 2
j j −n 1 − + |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) n=0 1 j −n ·log + |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1)
D(ρ ab ) = F log
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES
−
j −n 1 − |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) 1 j −n ·log − |(F (2j + 1) − j )| . 2j + 1 j (j + 1)(2j + 1)
169
j n=0
(12)
evaluate the one-way deficit of ρ ab , we calculate the eigenvalues of To ab
k ρ k = p0 ρ0 + p1 ρ1 . Since 2j1+1 I − M commutes with 2j1+1 I + M, by k 1 ± using (19) and (20) in [30], we have the eigenvalues of
k ρ ab k , μ± n = 2 λn , k
each with algebraic multiplicity two. As the eigenvalues do not depend on the measurement parameters, the minimum of entropy of the post-measurement states does not require any optimization over the projective measurements, min S { k }
k
j −n 1 ± |(F (2j + 1) − j )|
k ρ k = − (2j + 1) j (j + 1)(2j + 1) n=0 1 j −n · log ± |(F (2j + 1) − j )| . (13) 2(2j + 1) 2j (j + 1)(2j + 1) ab
j
From Eqs. (1), (4) and (13), the one-way deficit of the state is given by → (ρ ab ) = min S
k ρ ab k − S(ρ ab ) { k }
k
1−F F + (1 − F ) log = F log 2j 2j + 2
j j −n 1 − + |(F (2j + 1) − j )| (2j + 1) j (j + 1)(2j + 1) n=0 1 1 j −n · log + |(F (2j + 1) − j )| 2 (2j + 1) j (j + 1)(2j + 1)
j j −n 1 − − |(F (2j + 1) − j )| (2j + 1) j (j + 1)(2j + 1) n=0 1 1 j −n · log − |(F (2j + 1) − j )| 2 (2j + 1) j (j + 1)(2j + 1) 1−F 2 F + ( j + 1) = F log + (1 − F )log 2j 2j + 2 2j + 1
j j −n 1 − + |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) n=0
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Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG
1 j −n ·log + |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1)
j j −n 1 − − |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) n=0 1 j −n ·log − |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) ⎧ ⎨ D(ρ ab ), j is a half-integer, = 1 ⎩ D(ρ ab ) + , j is an integer, d
(14)
where d = 2j + 1. Especially, when j = 21 , the state becomes the 2 × 2 Werner state, and the one-way deficit is equal to the quantum discord, which is consistent with the result obtained in [30]. In Fig. 1 we show that quantum correlation is a function of F for different spin j . For half-integer j , we observe that as j increases, the one-way deficit increases for small F and decreases for large F . But for integer j , the one-way deficit decreases as j increases, see Fig. 1a. For high-dimensional system (large j ), the one-way deficit becomes symmetric around the point F = 12 where → (ρ ab ) vanishes, see Fig. 1b. Maximum of the one-way deficit is attained at F = 1 and at F = 0 for systems of any dimension. From Eq. (14), one can see an interesting fact: for half-integer j , the local density matrices of the SU(2) invariant states are ρ a = 2j1+1 I2j +1 and ρ b = 12 I2 , where I2j +1 and I2 are the identity matrix of size 2j + 1 and 2, respectively. The one-way deficit is equal to the quantum discord. It is consistent with the fact that if the local density matrices of a state are maximally mixed, the one-way work-deficit and the discord of that state are the same, see Fig. 1c. For integer j , the difference between the one-way deficit and the quantum discord is d1 , see Fig. 1c, and the difference tends to be small for large j , see Fig. 1d. We now compare the one-way deficit with the entanglement of formation (EoF). The EoF for a spin- 12 and a spin-j SU(2) invariant state ρ ab is given by [35], ⎧ 2j ⎪ ⎪ 0, F ∈ 0, ⎪ ⎨ 2j + 1 EoF(ρ ab ) = (15) √ 2 ⎪ 2j 1 ⎪ ⎪ ,1 , F − 2j (1 − F ) , F ∈ ⎩H 2j + 1 2j + 1 where H (x) = −xlogx − (1 − x)log(1 − x) is the binary entropy. It is shown that EoF becomes an upper bound for quantum discord for d × d Werner states [41]. However, we can see that the one-way deficit always remains larger than EoF for half-integer j . The difference between EoF and → (ρ ab ) increases if j → ∞ [39]. But for integer j , the difference decreases if j → ∞, see Fig. 2a and b. It
171
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES 2.0
(a)
OWD
1.5
1.0
j=1 j=2 j=3
0.5
j=0.5 j=1.5 j=2.5 0.0
0.0
0.2
0.4
0.6
0.8
1.0
F
1.0
(b)
0.8
OWD
0.6
0.4
0.2 j=300 j=299.5 0.0
0.0
0.2
0.4
0.6
0.8
1.0
F
Fig. 1. (Color online) quantum correlations of the bipartite state composed of a spin-j and a spin- 12 vs F . (a) one-way deficit of j = 1 (green line), j = 2 (yellow line), j = 3 (black line) and j = 12 (red line), j = 32 (blue line), j = 52 (gray line), (b) one-way deficit of j = 300 (red line) and j = 599 2 (blue line), (c) comparison of one-way deficit (solid line) and quantum discord(dotted line) of j = 1 (green line), j = 2 (black line ) and j = 12 (red line), j = 32 (blue line), (d) comparison of one-way deficit( red solid line) and quantum discord (blue dotted line) of j = 50.
172
Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG 2.0
(c)
___ .....
Correlations
1.5
OWD QD
1.0 j=1 j=2 j=1 0.5
j=0.5 j=2 j=1.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
F
1.0
(d)
___ 0.8
.....
OWD QD
Correlations
j=50 0.6
0.4
0.2
0.0
0.0
0.2
0.4 F
Fig. 1. cont.
should be noted that if j → ∞, the state ρ ab becomes separable while its one-way deficit remains finite. In the region in which EoF is zero, the one-way deficit survives.
173
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES 2.0
(a)
___
OWD
..... EoF
Correlations
1.5
j=1
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
F 2.0
(b)
___
OWD
..... EoF
1.5
Correlations
j=5
1.0
0.5
0.0
0.0
0.2
0.4 F
Fig. 2. (Color online) one-way deficit (red solid line) and EoF (blue dotted line) vs. F : (a) j = 1, (b) j = 5.
3.
Conclusion
We have analytically calculated the one-way deficit of SU(2) invariant states consisting of a spin-j and a spin- 12 subsystems, with the measurement on the spin- 12 subsystem. By comparing the one-way deficit with the quantum discord of these states we have shown that the one-way deficit is equal to the quantum
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Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG
discord for half-integer j , and the one-way deficit is larger than the quantum discord for integer j . We have also compared our results on one-way deficit with the quantum entanglement EoF. It is shown that for large j , the one-way deficit remains significantly larger than EoF. Moreover, the maximal value of one-way deficit decreases with the increasing system size. As there is abundance of SU(2) invariant states in real physical systems, our results can be used in quantum protocols that rely on one-way deficit. Acknowledgments This work is supported by NSFC11275131, NSFC11305105 and KZ201210028032. REFERENCES [1] C. H. Bennett and S. Wiesner: Phys. Rev. Lett. 69 (1992), 2881. [2] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres and W. K. Wootters: Phys. Rev. Lett. 70 (1993), 1895. [3] A. K. Ekert: Phys. Rev. Lett. 67 (1991), 661. [4] A. K. Pati: Phys. Rev. A 63 (2000); C, 014302. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal and W. K. Wootters: Phys. Rev. Lett. 87 (2001), 077902. [5] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin and W. K. Wootters: Phys. Rev. A 59 (1999), 1070. [6] L. Henderson and V. Vedral: J. Phys. A 34 (2001), 6899; H. Ollivier and W. H. Zurek: Phys. Rev. Lett. 88 (2002), 017901. [7] K. Modi, T. Paterek, W. Son, V. Vedral and M. Williamson: Phys. Rev. Lett. 104 (2010), 080501. [8] R. Auccaise, J. Maziero, L. C. C´eleri, D. O. Soares-Pinto, E. R. deAzevedo, T. J. Bonagamba, R. S. Sarthour, I. S. Oliveira and R. M. Serra: Phys. Rev. Lett. 107 (2010), 070501. [9] G. L. Giorgi, B. Bellomo, F. Galve and R. Zambrini: Phys. Rev. Lett. 107 (2011), 190501. [10] A. Streltsov, G. Adesso, M. Piani and D. Bruß: Phys. Rev. Lett. 109 (2012), 050503. [11] K. Modi, A. Brodutch, H. Cable, T. Paterek and V. Vedral: Rev. Mod. Phys. 84 (2012), 1655. [12] S. Luo: Phys. Rev. A 77 (2008), 042303. [13] M. Ali, A. R. P. Rau and G. Alber: Phys. Rev. A 81 (2010), 042105. [14] B. Li, Z. X. Wang and S. M. Fei: Phys. Rev. A 83 (2011), 022321. [15] Q. Chen, C. Zhang, S. Yu, X. X. Yi and C. H. Oh: Phys. Rev. A 84 (2011), 042313. [16] M. Shi, C. Sun, F. Jiang, X. Yan and J. Du: Phys. Rev. A 85 (2012), 064104. [17] S. Vinjanampathy, A. R. P. Rau: J. Phys. A 45 (2012), 095303. [18] J. Oppenheim, M. Horodecki, P. Horodecki and R. Horodecki: Phys. Rev. Lett. 89 (2002), 180402. [19] M. Horodecki, K. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen(De) and U. Sen: Phys. Rev. Lett. 90 (2003), 100402. [20] M. Piani, P. Horodecki and R. Horodecki: Phys. Rev. Lett. 100 (2008), 090502. [21] S. Wu, U. V. Poulsen and K. Mølmer: Phys. Rev. A 80 (2009), 032319. [22] K. Modi, T. Paterek, W. Son, V. Vedral and M. Williamson: Phys. Rev. Lett. 104 (2010), 080501. [23] S. Luo and S. Fu: Phys. Rev. A 82 (2010), 034302. [24] B. Dakic, V. Vedral and C. Brukner: Phys. Rev. Lett. 105 (2010), 190502. [25] G. Adesso and A. Datta: Phys. Rev. Lett. 105 (2010), 030501. [26] P. Giorda and M. G. A. Paris: Phys. Rev. Lett. 105 (2010), 020503. [27] A. Streltsov, H. Kampermann and D. Bruß: Phys. Rev. Lett. 108 (2012), 250501. [28] T. K. Chuan, J. Maillard, K. Modi, T. Paterek, M. Paternostro and M. Piani: Phys. Rev. Lett. 109 (2012), 070501. [29] M. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen(De), U. Sen and B. Synak: Phys. Rev. A 71 (2005), 062307.
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]
Y. K. Wang, T. Ma, B. Li, Z. X. Wang: Commun. Theor. Phys. 59 (2013), 540. J. Schliemann: Phys. Rev. A 68 (2003), 012309. J. Schliemann: Phys. Rev. A 72 (2005), 012307. G. A. Durkin, C. Simon, J. Eisert and D. Bouwmeester: Phys. Rev. A 70 (2004), 062305. H. P. Breuer: Phys. Rev. A 71 (2005), 062330. H. P. Breuer: J. Phys. A 38 (2005), 9019. Z. Wang and Z. X. Wang: Phys. Lett. A 372 (2008), 7033. K. K. Manne and C. M. Caves: Quantum. Inf. Comp. 8 (2008), 0295. K. G. H. Vollbrecht and R. F. Werner: Phys. Rev. A 64 (2001), 062307. B. Cakmak and Z. Gedik: J. Phys. A 46 (2013), 465302. R. Shankar: Principles of Quantum Mechanics, Plenum Press, New York, 1994. E. Chitambar: Phys. Rev. A 86 (2012), 032110.
175
REPORTS ON MATHEMATICAL PHYSICS
No. 2
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES YAO -K UN WANG1,2 , T ENG M A1 , S HAO -M ING F EI1,3 and Z HI -X I WANG1 1 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China 2 College of Mathematics, Tonghua Normal University, Tonghua 134001, China 3 Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
(e-mails: [email protected] [email protected] [email protected] [email protected]) (Received June 12, 2013 – Revised November 12, 2013) We calculate analytically the one-way deficit for systems consisting of spin-j and spin- 12 subsystems with SU(2) symmetry. Comparing our results with the quantum discord of SU(2) invariant states, we show that the one-way deficit is equal to the quantum discord for half-integer j , and is larger than the quantum discord for integer j . Moreover, we also compare the one-way deficit with entanglement of formation. The quantum entanglement tends to zero if j increases, while the one-way deficit can remain significantly large. Keywords: one-way deficit, SU(2) invariant state, quantum discord.
1.
Introduction Quantum entanglement plays significant roles in the field of quantum information and quantum computation such as super-dense coding [1], teleportation [2], quantum cryptography [3], remote-state preparation [4]. Quantum correlations other than quantum entanglement also attract much attention recently [5–11]. Among them, quantum discord introduced by Oliver and Zurek and independently by Henderson and Vedral [6] is a widely accepted quantity. Quantum discord, a measure which quantifies discrepancy between the quantum mutual information and the maximal classical information, can be present in separable mixed quantum states. Following this discovery, much work has been done in investigating the properties and behaviour of quantum discord in various systems. Since complicated optimization procedure is involved in calculating quantum discord, one has no analytical formulae of quantum discord even for two-qubit quantum systems [12–17]. Other significant nonclassical correlations besides entanglement and quantum discord, for example, the quantum deficit [18, 19], measurement-induced disturbance [12], symmetric discord [20, 21], relative entropy of discord and dissonance [22], geometric discord [23, 24], and continuous-variable discord [25, 26] have been studied recently. The work deficit [18] proposed to characterize quantum correlations in terms of entropy production and work extraction by Oppenheim et al. is the first operational approach to connect the quantum correlations theory with quantum thermodynamics. Recently, Streltsov et al. [27, 28] has revealed that the one-way deficit plays [165]
166
Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG
an important role in quantum correlations as a resource for the distribution of entanglement. The one-way deficit of a bipartite quantum state ρ is defined by [29]
k ρ k − S(ρ), (1) → (ρ) = min S { k }
k
where { k } are the projective measurements and S is the von Neumann entropy. The one-way deficit and quantum discord have similar minimum forms but they are different kinds of quantum correlations. We have obtained an analytical formula of one-way deficit for some well-known states such as Bell-diagonal states [30]. In this paper, we endeavoured to calculate the one-way deficit of bipartite SU(2) invariant states consisting of a spin-j and a spin- 12 subsystems. SU(2)-invariant density matrices of two spins S1 and S2 are defined to be invariant under U1 ⊗ U2 , that is, U1 ⊗ U2 ρU1† ⊗ U2† = ρ, where Ua = exp(i η · Sa ), a = 1, 2, are the usual rotation operator representations of SU(2) with real parameters η and h¯ = 1 [31, 32]. Those SU(2)-invariant states commute with all components of the total spin J = S1 + S2 . In real physical systems, SU(2)-invariant states arise from thermal equilibrium states of spin systems described by SU(2) invariant Hamiltonian [33]. The state space structure and entanglement of SO(3)-invariant bipartite quantum systems have been analyzed in the literature [34, 35]. For SU(2) invariant quantum spin systems composed of spin-S and spin- 12 , negativity is shown to be necessary and sufficient for separability [31, 32], and the relative entropy of entanglement has been analytically calculated [36] for (2j + 1) × 2 and ((2j + 1) × 3)-dimensional systems. Furthermore, the entanglement of formation (EoF), I-concurrence, I-tangle and convex-roof-extended negativity of the SU(2)-invariant states of a spin-j and spin- 12 [37] have been analytically calculated by using the approach of [38]. The quantum discord for SU(2)-invariant states composed of spin-j and spin- 21 systems has been analytically calculated in [39]. As an SU(2)-invariant state commutes with all components of J, ρ has the general form [31] S J 1 +S2 A(J ) ρ= |J, J z 00 J, J z | , (2) 2J + 1 J =|S1 −S2 | J z =−J where the constants A(J ) ≥ 0, J A(J ) = 1, |J, J z 0 denotes a state of the total spin J and the z-component J z . We consider the case with S1 of arbitrary length S and S2 of length 12 . Let S = j , J z = m. A general SU(2)-invariant density matrix has the form j − 12 1 F 1 ab ρ = j − 2 , m j − 2 , m 2j 1 m=−j + 2
1−F + 2(j + 1)
j + 12
j + 1 , m j + 1 , m , 2 2 1
m=−j − 2
(3)
167
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES
where F ∈ [0, 1], and F is a function of temperature in thermal equilibrium. ρ ab is a ((2j +1)×2)-bipartite state. It has two eigenvalues λ1 = F /2j and λ2 = (1−F )/(2j + 2) with degeneracies 2j and 2j +2, respectively [39]. The entropy of ρ ab is given by S(ρ ab ) = −F log
1−F F − (1 − F ) log . 2j 2j + 2
(4)
As the eigenstates of the total spin can be given by the Clebsch–Gordan coefficients [40] in the coupling of a spin-j to spin- 12 , j ± 1 , m 2 1 1 1 j + 12 ± m j + 12 ∓ m 1 1 j, m − , j, m + , − 1 , (5) ⊗ + ⊗ =± 2 2 2 2 2j + 1 2 2j + 1 2 the density matrix (3) can be written in the product basis form [39], F ρ ab = 2j
j − 12
2 a−
m=−j + 12
m − 1 m − 2
1 1 1 ⊗ 2 2 2
1 1 1 1 1 m+ ⊗ − + m + m− +a− b− m − 2 2 2 2 2 1 1 1 1 2 m + ⊗ − − m + +b− 2 2 2 2 j + 12 1−F 1 1 1 1 2 + m− ⊗ a+ m − 2(j + 1) 2 2 2 2 1
1 1 1 ⊗ − 2 2 2
1 1 1 1 1 m+ ⊗ − + m + m− +a+ b+ m − 2 2 2 2 2 1 1 1 1 2 m + ⊗ − − , +b+ m + 2 2 2 2
1 1 1 ⊗ − 2 2 2
m=−j − 2
where
j + 12 ± m a± = ± 2j + 1
and
b± =
(6)
j + 12 ∓ m . 2j + 1
When j = 12 , the state ρ ab turns out to be a 2 × 2 Werner state, I ρ = (1 − c) + c|ψ − ψ − |, 4 with |ψ − =
√1 (|01
2
− |10 ).
c=
4F − 1 , 3
(7)
168 2.
Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG
The main result Any von Neumann measurement on the spin- 12 subsystem can be written as Bk = V k V † ,
k = 0, 1,
(8)
where k = |k k|, |k is the computational basis, V = tI + i y · σ ∈ SU(2), σ = (σ1 , σ2 , σ3 ) with σ1 , σ2 , σ3 being the Pauli matrices, t ∈ R and y = (y1 , y2 , y3 ) ∈ R3 , t 2 + y12 + y22 + y32 = 1. Set j m(2Fj + F − j ) M= |m m| z3 j (j + 1)(2j + 1) m=−j √ j (j + 1) − m(m + 1)(2Fj + F − j ) +(z1 + iz2 ) |m m + 1| 2j (j + 1)(2j + 1) √ j (j + 1) − m(m + 1)(2Fj + F − j ) |m + 1 m| , +(z1 − iz2 ) 2j (j + 1)(2j + 1) where z1 = 2(−ty2 + y1 y3 ), z2 = 2(ty1 + y2 y3 ), z3 = t 2 + y32 − y12 − y22 with z12 + z22 + z32 = 1. After the measurement, one has an ensemble of post-measurement states {ρk , pk } with p0 = p1 = 12 and the corresponding post-measurement states [39] j 1 1 ρ0 = |m m| − M ⊗ V 0 V † = I − M ⊗ V 0 V † , (9) 2j + 1 m=−j 2j + 1 and
ρ1 =
j 1 1 † I + M ⊗ V 1 V † . (10) |m m| + M ⊗ V 1 V = 2j + 1 m=−j 2j + 1
The eigenvalues of ρ0 , ρ1 [39] are λ± n =
j −n 1 ± |(F (2j + 1) − j )|, 2j + 1 j (j + 1)(2j + 1)
(11)
where n = 0, . . . , j , and j denotes the largest integer that is less or equal to j . Obviously, eigenvalues are independent of the measurement. Analytical expression for quantum discord of ρ ab has been obtained in [39], 1−F F + (1 − F )log +1 2j 2j + 2
j j −n 1 − + |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) n=0 1 j −n ·log + |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1)
D(ρ ab ) = F log
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES
−
j −n 1 − |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) 1 j −n ·log − |(F (2j + 1) − j )| . 2j + 1 j (j + 1)(2j + 1)
169
j n=0
(12)
evaluate the one-way deficit of ρ ab , we calculate the eigenvalues of To ab
k ρ k = p0 ρ0 + p1 ρ1 . Since 2j1+1 I − M commutes with 2j1+1 I + M, by k 1 ± using (19) and (20) in [30], we have the eigenvalues of
k ρ ab k , μ± n = 2 λn , k
each with algebraic multiplicity two. As the eigenvalues do not depend on the measurement parameters, the minimum of entropy of the post-measurement states does not require any optimization over the projective measurements, min S { k }
k
j −n 1 ± |(F (2j + 1) − j )|
k ρ k = − (2j + 1) j (j + 1)(2j + 1) n=0 1 j −n · log ± |(F (2j + 1) − j )| . (13) 2(2j + 1) 2j (j + 1)(2j + 1) ab
j
From Eqs. (1), (4) and (13), the one-way deficit of the state is given by → (ρ ab ) = min S
k ρ ab k − S(ρ ab ) { k }
k
1−F F + (1 − F ) log = F log 2j 2j + 2
j j −n 1 − + |(F (2j + 1) − j )| (2j + 1) j (j + 1)(2j + 1) n=0 1 1 j −n · log + |(F (2j + 1) − j )| 2 (2j + 1) j (j + 1)(2j + 1)
j j −n 1 − − |(F (2j + 1) − j )| (2j + 1) j (j + 1)(2j + 1) n=0 1 1 j −n · log − |(F (2j + 1) − j )| 2 (2j + 1) j (j + 1)(2j + 1) 1−F 2 F + ( j + 1) = F log + (1 − F )log 2j 2j + 2 2j + 1
j j −n 1 − + |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) n=0
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Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG
1 j −n ·log + |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1)
j j −n 1 − − |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) n=0 1 j −n ·log − |(F (2j + 1) − j )| 2j + 1 j (j + 1)(2j + 1) ⎧ ⎨ D(ρ ab ), j is a half-integer, = 1 ⎩ D(ρ ab ) + , j is an integer, d
(14)
where d = 2j + 1. Especially, when j = 21 , the state becomes the 2 × 2 Werner state, and the one-way deficit is equal to the quantum discord, which is consistent with the result obtained in [30]. In Fig. 1 we show that quantum correlation is a function of F for different spin j . For half-integer j , we observe that as j increases, the one-way deficit increases for small F and decreases for large F . But for integer j , the one-way deficit decreases as j increases, see Fig. 1a. For high-dimensional system (large j ), the one-way deficit becomes symmetric around the point F = 12 where → (ρ ab ) vanishes, see Fig. 1b. Maximum of the one-way deficit is attained at F = 1 and at F = 0 for systems of any dimension. From Eq. (14), one can see an interesting fact: for half-integer j , the local density matrices of the SU(2) invariant states are ρ a = 2j1+1 I2j +1 and ρ b = 12 I2 , where I2j +1 and I2 are the identity matrix of size 2j + 1 and 2, respectively. The one-way deficit is equal to the quantum discord. It is consistent with the fact that if the local density matrices of a state are maximally mixed, the one-way work-deficit and the discord of that state are the same, see Fig. 1c. For integer j , the difference between the one-way deficit and the quantum discord is d1 , see Fig. 1c, and the difference tends to be small for large j , see Fig. 1d. We now compare the one-way deficit with the entanglement of formation (EoF). The EoF for a spin- 12 and a spin-j SU(2) invariant state ρ ab is given by [35], ⎧ 2j ⎪ ⎪ 0, F ∈ 0, ⎪ ⎨ 2j + 1 EoF(ρ ab ) = (15) √ 2 ⎪ 2j 1 ⎪ ⎪ ,1 , F − 2j (1 − F ) , F ∈ ⎩H 2j + 1 2j + 1 where H (x) = −xlogx − (1 − x)log(1 − x) is the binary entropy. It is shown that EoF becomes an upper bound for quantum discord for d × d Werner states [41]. However, we can see that the one-way deficit always remains larger than EoF for half-integer j . The difference between EoF and → (ρ ab ) increases if j → ∞ [39]. But for integer j , the difference decreases if j → ∞, see Fig. 2a and b. It
171
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES 2.0
(a)
OWD
1.5
1.0
j=1 j=2 j=3
0.5
j=0.5 j=1.5 j=2.5 0.0
0.0
0.2
0.4
0.6
0.8
1.0
F
1.0
(b)
0.8
OWD
0.6
0.4
0.2 j=300 j=299.5 0.0
0.0
0.2
0.4
0.6
0.8
1.0
F
Fig. 1. (Color online) quantum correlations of the bipartite state composed of a spin-j and a spin- 12 vs F . (a) one-way deficit of j = 1 (green line), j = 2 (yellow line), j = 3 (black line) and j = 12 (red line), j = 32 (blue line), j = 52 (gray line), (b) one-way deficit of j = 300 (red line) and j = 599 2 (blue line), (c) comparison of one-way deficit (solid line) and quantum discord(dotted line) of j = 1 (green line), j = 2 (black line ) and j = 12 (red line), j = 32 (blue line), (d) comparison of one-way deficit( red solid line) and quantum discord (blue dotted line) of j = 50.
172
Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG 2.0
(c)
___ .....
Correlations
1.5
OWD QD
1.0 j=1 j=2 j=1 0.5
j=0.5 j=2 j=1.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
F
1.0
(d)
___ 0.8
.....
OWD QD
Correlations
j=50 0.6
0.4
0.2
0.0
0.0
0.2
0.4 F
Fig. 1. cont.
should be noted that if j → ∞, the state ρ ab becomes separable while its one-way deficit remains finite. In the region in which EoF is zero, the one-way deficit survives.
173
ONE-WAY DEFICIT OF SU(2) INVARIANT STATES 2.0
(a)
___
OWD
..... EoF
Correlations
1.5
j=1
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
F 2.0
(b)
___
OWD
..... EoF
1.5
Correlations
j=5
1.0
0.5
0.0
0.0
0.2
0.4 F
Fig. 2. (Color online) one-way deficit (red solid line) and EoF (blue dotted line) vs. F : (a) j = 1, (b) j = 5.
3.
Conclusion
We have analytically calculated the one-way deficit of SU(2) invariant states consisting of a spin-j and a spin- 12 subsystems, with the measurement on the spin- 12 subsystem. By comparing the one-way deficit with the quantum discord of these states we have shown that the one-way deficit is equal to the quantum
174
Y. K. WANG, T. MA, S. M. FEI and Z. X. WANG
discord for half-integer j , and the one-way deficit is larger than the quantum discord for integer j . We have also compared our results on one-way deficit with the quantum entanglement EoF. It is shown that for large j , the one-way deficit remains significantly larger than EoF. Moreover, the maximal value of one-way deficit decreases with the increasing system size. As there is abundance of SU(2) invariant states in real physical systems, our results can be used in quantum protocols that rely on one-way deficit. Acknowledgments This work is supported by NSFC11275131, NSFC11305105 and KZ201210028032. REFERENCES [1] C. H. Bennett and S. Wiesner: Phys. Rev. Lett. 69 (1992), 2881. [2] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres and W. K. Wootters: Phys. Rev. Lett. 70 (1993), 1895. [3] A. K. Ekert: Phys. Rev. Lett. 67 (1991), 661. [4] A. K. Pati: Phys. Rev. A 63 (2000); C, 014302. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal and W. K. Wootters: Phys. Rev. Lett. 87 (2001), 077902. [5] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin and W. K. Wootters: Phys. Rev. A 59 (1999), 1070. [6] L. Henderson and V. Vedral: J. Phys. A 34 (2001), 6899; H. Ollivier and W. H. Zurek: Phys. Rev. Lett. 88 (2002), 017901. [7] K. Modi, T. Paterek, W. Son, V. Vedral and M. Williamson: Phys. Rev. Lett. 104 (2010), 080501. [8] R. Auccaise, J. Maziero, L. C. C´eleri, D. O. Soares-Pinto, E. R. deAzevedo, T. J. Bonagamba, R. S. Sarthour, I. S. Oliveira and R. M. Serra: Phys. Rev. Lett. 107 (2010), 070501. [9] G. L. Giorgi, B. Bellomo, F. Galve and R. Zambrini: Phys. Rev. Lett. 107 (2011), 190501. [10] A. Streltsov, G. Adesso, M. Piani and D. Bruß: Phys. Rev. Lett. 109 (2012), 050503. [11] K. Modi, A. Brodutch, H. Cable, T. Paterek and V. Vedral: Rev. Mod. Phys. 84 (2012), 1655. [12] S. Luo: Phys. Rev. A 77 (2008), 042303. [13] M. Ali, A. R. P. Rau and G. Alber: Phys. Rev. A 81 (2010), 042105. [14] B. Li, Z. X. Wang and S. M. Fei: Phys. Rev. A 83 (2011), 022321. [15] Q. Chen, C. Zhang, S. Yu, X. X. Yi and C. H. Oh: Phys. Rev. A 84 (2011), 042313. [16] M. Shi, C. Sun, F. Jiang, X. Yan and J. Du: Phys. Rev. A 85 (2012), 064104. [17] S. Vinjanampathy, A. R. P. Rau: J. Phys. A 45 (2012), 095303. [18] J. Oppenheim, M. Horodecki, P. Horodecki and R. Horodecki: Phys. Rev. Lett. 89 (2002), 180402. [19] M. Horodecki, K. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen(De) and U. Sen: Phys. Rev. Lett. 90 (2003), 100402. [20] M. Piani, P. Horodecki and R. Horodecki: Phys. Rev. Lett. 100 (2008), 090502. [21] S. Wu, U. V. Poulsen and K. Mølmer: Phys. Rev. A 80 (2009), 032319. [22] K. Modi, T. Paterek, W. Son, V. Vedral and M. Williamson: Phys. Rev. Lett. 104 (2010), 080501. [23] S. Luo and S. Fu: Phys. Rev. A 82 (2010), 034302. [24] B. Dakic, V. Vedral and C. Brukner: Phys. Rev. Lett. 105 (2010), 190502. [25] G. Adesso and A. Datta: Phys. Rev. Lett. 105 (2010), 030501. [26] P. Giorda and M. G. A. Paris: Phys. Rev. Lett. 105 (2010), 020503. [27] A. Streltsov, H. Kampermann and D. Bruß: Phys. Rev. Lett. 108 (2012), 250501. [28] T. K. Chuan, J. Maillard, K. Modi, T. Paterek, M. Paternostro and M. Piani: Phys. Rev. Lett. 109 (2012), 070501. [29] M. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen(De), U. Sen and B. Synak: Phys. Rev. A 71 (2005), 062307.
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